220.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

3 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 220

References:

erdosproblems.com/220
[Er40] Erdős, P., The difference of consecutive primes. Duke Math. J. (1940), 438--441.
[Er73] Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138.
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
[MoVa86] Montgomery, H. L. and Vaughan, R. C., On the distribution of reduced residues. Ann. of Math. (2) (1986), 311-333.
[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.

Imports (1)

FormalConjecturesUtil

def Erdos220.sumSquaredGaps The sum of the squared differences of adjacent entries of a list of natural numbers line 42 source
Source documentation
The sum of the squared differences of adjacent entries of a list of natural numbers.

standard axioms: propext what this means
Statement uses: List, Nat
2 instances, coercions and auxiliary constants left out.
```
def sumSquaredGaps : List ℕ → ℕ
  | a :: b :: rest => (b - a) ^ 2 + sumSquaredGaps (b :: rest)
  | _ => 0

/-- The reduced residues `1 ≤ m < n` with `(m, n) = 1`, in increasing order. -/
```
def Erdos220.sortedTotatives The reduced residues 1 ≤ m < n with (m, n) = 1, in increasing order line 47 source
Source documentation
The reduced residues 1 ≤ m < n with (m, n) = 1, in increasing order.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, List
Proof uses: Finset.sort, Finset.filter, Nat.Coprime, Finset.Ico, OfNat.ofNat, LE.le
10 instances, coercions and auxiliary constants left out.
```
def sortedTotatives (n : ℕ) : List ℕ :=
…
```
theorem Erdos220.erdos_220 sorry line 63 source
depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, LT.lt, OfNat.ofNat, Nat, LE.le, Erdos220.sumSquaredGaps, Erdos220.sortedTotatives, HDiv.hDiv, HMul.hMul, HPow.hPow, Nat.totient
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.